Real Polarizable Hodge Structures Arising from Foliations

We construct real polarizable Hodge structures on the reduced leafwisecohomology of Kähler–Riemann foliations by complex manifolds. As inthe classical case one obtains a hard Lefschetz theorem for thiscohomology. Serre's Kählerin analogue of the Weil conjectures carriesover as well. Generalizing a construction of Looijenga and Lunts oneobtains possibly infinite-dimensional Lie algebras attached toKähler–Riemann foliations.